1. CHAPTER 10 HEAT TRANSFER IN LIVING TISSUE 10.1 Introduction · Examples
·    Hyperthermia
·    Cryosurgery
·    Skin burns
·    Frost bite
·    Body thermal regulation
·    Modeling
Modeling heat transfer in living tissue requires
the formulation of a special heat equation 1

2. 10.2 Vascular Architecture and Blood Flow
Key features
(1) Blood perfused tissue
(2) Vascular architecture
(3) Variation in blood flow rate and tissue properties
10.2 Vascular Architecture and Blood Flow
Vessels
Artery/vein
Aorta/vena cava
Supply artery/vein
Primary vessels
Secondary vessels
Arterioles/venules
Capillaries 2

3. 10.3 Blood Temperature Variation
Blood leaves heart at
Equilibration with tissue: prior to arterioles and capillaries
Metabolic heat is removed from blood near skin
Blood mixing from various sources brings temperature to 3

4. 10.4 Mathematical Modeling of Vessels-Tissue Heat Transfer
Pennes Bioheat Equation (1948)
(a) Formulation
Assumptions:
(1) Equilibration Site:
Arterioles, capillaries & venules
(2) Blood Perfusion:
Neglects flow directionality. i.e.
isotropic blood flow
(3) Vascular Architecture:
No influence
(4) Blood Temperature: 4

5. Treat energy exchange due to blood perfusion as energy generation
Conservation of energy for the
element shown in Fig. 10.3:
Treat energy exchange due to blood perfusion as energy generation
Let = b q
net rate of energy added by the blood per unit
volume of tissue = m q
rate of metabolic energy production per unit
volume of tissue dz dy dx q E m b g ) ( + =
&
(a) 5

6. (10.3) 6

7. Cartesian coordinates:
cylindrical coordinates:
spherical coordinates: 7

8. (b) Shortcomings of the Pennes equation
Notes on eq. (10.3):
(10.3)
This is known as the Pennes Bioheat equation
The blood perfusion term is mathematically identical to surface convection in fins, eqs. (2.5), (2.19), (2.23) and (2.24)
(3) The same effect is observed in porous fins with coolant flow (see problems 5.12, 5.17, and 5.18)
(b) Shortcomings of the Pennes equation
Equilibration Site:
(1)
Does not occur in the capillaries 8

9. · · · Occurs in the thermally significant pre - arteriole and post
venule vessels (dia. 70
500 m )
Thermally significant vessels : 1
> L e e L =
Equilibration length
: distance blood travels for
its temperatur
e to equilibrate with tissue
Blood Perfusion:
(2)
·  Perfusion in not isotropic
·  Directionality is important in energy interchange
Vascular Architecture :
(3)
Local vascular geometry not accounted for
Neglects artery-vein countercurrent heat exchange
Neglects influence of nearby large vessels 9

10. (c) Applicability (4) Blood Temperature:
· Blood does not reach tissue at body core temperature
· Blood does not leave tissue at local temperature T
(c) Applicability
· Surprisingly successful, wide applications
· Reasonable agreement with some experiments 10

11. Example 10.1: Temperature Distribution in the Forearm
Model forearm as a cylinder
Blood perfusion rate b w
&
Metabolic heat production m q
Convection at the surface
Heat transfer coefficient is h
Ambient temperature is T
Use Pennes bioheat equation to
determine the 1 - D
temperature distribution
(1) Observations 11

12. (2) Origin and Coordinates. See Fig. 10.4 (3) Formulation
Arm is modeled as a cylinder with uniform energy
generation
·   Heat is conduction to skin and removed by convection
·   In general, temperature distribution is 3-D
(2) Origin and Coordinates. See Fig. 10.4
(3) Formulation
(i) Assumptions
(1)          Steady state
(2)          Forearm is modeled as a constant radius cylinder
(3)          Bone and tissue have the same uniform properties
(4)          Uniform metabolic heat
(5)          Uniform blood perfusion
(6)          No variation in the angular direction
(7)          Negligible axial conduction 12

13. (ii) Governing Equations
(8) Skin layer is neglected
(9) Pennes bioheat equation is applicable
(ii) Governing Equations
Pennes equation (10.3) for 1-D steady state radial heat transfer
(iii) Boundary Conditions:
(4) Solution 13

14. Rewrite (a) in dimensionless form. Define
(d) into (a)
Define
(f) and (g) into (e) 14

15. Bi is the Biot number The boundary conditions become
Homogeneous part of (h) is a Bessel differential equation.
The solution is 15

16. (5) Checking Boundary conditions give (m) into (k)
Dimensional check: Bi,and are dimensionless. The arguments of the Bessel functions are dimensionless.
Dimensional check:
Limiting check: If no heat is removed (),arm reaches a uniform temperature . All metabolic heat is transferred to the blood. Conservation of energy for the blood:
Limiting check: 16

17. Solve for
which agrees with (o)
(6) Comments 17

18. Chen-Holmes Equation
First to show that equilibration occurs prior to reaching
the arterioles
Accounts for blood directionality
Accounts for vascular geometry
The Pennes equation is modified to: 18

19. NOTE: 19

20. 10.4.3 Three-Temperature Model for Peripheral Tissue
Limitations
(1)
Vessel diameter m
300
<
(2) 6 .
< L e
(3)
Requires detailed knowledge of the vascular network
and blood perfusion
Three-Temperature Model for Peripheral
Tissue
Rigorous Approach
Accounts for vasculature and blood flow directionality 20

21. Assign three temperature variables: (1) Arterial temperature
(2)Venous temperature
(3) Tissue temperature T
Identify three layers:
Intermediate layer:
porous media
Cutaneous layer: thin,
independently supplied by counter-current artery-vein
vessels called cutaneous plexus
Regulates surface heat flux 21

22. Formulation Consists of two regions:
(i) Thin layer near skin with negligible blood flow
(ii) Uniformly blood perfused layer (Pennes model)
Formulation
Seven equation:
3 for the deep layer
2 for the intermediate layer
2 for the cutaneous layer
Model is complex
Simplified form for the deep layer is presented in the next
section
Attention is focused on the cutaneous layer:
(i) Region 1, blood perfused. For 1-D steady state: 22

23. 10.4.3 Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue) ( 1 2 = - + T k w c dx d cb b
& r
(10.5) = 1 T
temperature variable in the lower layer = c T
temperature of blood supplying the cutaneous pelxus = cb w
&
cutane
ous layer blood perfusion rate = x
coordinate normal to skin surface
(ii) Region 2, pure conduction , for 1-D steady state: 2 = dx T d
(10.6)
Weinbaum-Jiji Simplified Bioheat
Equation for Peripheral Tissue
The 3 eqs. for , a T v
and
are replaced by one equation 23

24. Control Volume (a) Assumptions
Effect of vasculature and heat exchange between artery,
vein, and tissue are retained
Added simplification narrows applicability of result
Control Volume
Contains artery -
vein pairs
Countercurrent flow, v a T
Includes capillaries, arterioles
and venules
(a) Assumptions
(1)
Uniformly distributed blood
bleed -
off leaving artery is
equal to that returning to vein
(2)
Bleed -
off blood leaves artery at a T
and enters the vein at v 24

25. (b) Formulation (3) Artery and vein have the same radius
(4) Negligible axial conduction through vessels
(5) Equilibration length ratio 1 /
<< L e
(6) Tissue temperature T
is approximated by
(7) One-dimensional: blood vessels and temperature gradient
are in the same direction
(b) Formulation
Conservation of energy for tissue in control volume takes into
consideration:
(1) Conduction through tissue
(2) Energy exchange between vessels and tissue due to
capillary blood bleed-off from artery to vein 25

26. (3) Conduction between vessel pairs and tissue
Note: Conduction from artery to tissue not equal to conduction from the tissue to the vein (incomplete countercurrent exchange)
Conservation of energy for the artery, vein and tissue and conservation of mass for the artery and vein give ú û ù ê ë é + = 2 ) ( 1 u a c k n b
eff r p s
( 10.9) = a
vessel radius = n
number of vessel pairs crossing surface of control
volume per unit area = u
average blood velocity in countercurrent artery or vein 26

27. accounts for the effect of vascular geometry and blood perfusion · a ,
NOTE
eff k
accounts for the effect of vascular geometry and blood
perfusion a , s n
and u
depend on the vascular geometry
Conservation of mass gives u
in terms of inlet velocity o to
tissue layer
and the vascular geometry. Eq. (10.9) becomes 27

28. vessel radius at inlet to tissue layer, x = ) ( x V x = ) ( x V
dimensionless vascular geometry function
(independent of blood flow) = L x /
dimensionless distance = L
tiss
ue layer thickness = o u
blood velocity at inlet to tissue layer, x
NOTE: ) / 2 ( b o k u a c r
is independent of vascular geometry.
It represents the inlet Peclet number:
Eq. (10.12) into eq. (10.11)
Notes on
eff k : 28

29. Cutaneous layer: Use eqs. (10.5) and (10.6) ) ( = - + T k w c dx d & r
For the 3-D case, orientation of vessel pairs relative
to the direction of local tissue temperature gradient gives
rise to a tensor conductivity
(2) The second term on the right hand side of eqs. (10.11) and
(10.13) represents the enhancement in tissue conductivity
due to blood perfusion
Cutaneous layer:
Use eqs. (10.5) and (10.6) ) ( 1 2 = - + T k w c dx d cb b
& r
(10.12) 29

30. = number of arteries entering tissue layer per unit area
Eq. (10.12) into eq. (10.14)
Define R R =
total
rate of blood to the cutaneous layer to the
rate
of blood to the tissue layer 1 L
= is the thickness of the cutaneous layer
Eqs. (10.15) and (10.16) into (10.5) 30

31. (c) Limitation and Applicability
Results are compared
with 3 -
temperature model of
Section
Accurate tissue temperature
prediction for:
(1)
Vessel diameter < μm
200
(2)
Equilibration length ratio 2 . /
< L e
(3)
Peripheral tissue thickness < 2mm 31

32. Example 10.2: Temperature Distribution in Peripheral Tissue5 10 7 - 1 x ) ( V
10.7
Fig. =
eff k )] [ 2 Pe o +
Skin surface at s T
Blood supply temperature a T
Neglect blood flow
through cutaneous layer
vascular geometry is
described by ) ( x V 2 ) ( x C B A V + = 5 10 9 . 15 , 32 6 - = C
and B A
Use the Weinbaum-Jiji equation determine temperature distribution
(ii) Express results in dimensionless form: . 32

33. (1) Observations (iii) Plot showing effect of blood flow &
metabolic heat
(1)
Observations
Variation of k
with distance is known
Tissue can be modeled as a single
layer with variable
eff k
Metabolic heat is uniform
Temperature increases as blood perfusion
and/or metabolic heat are increased
(2) Origin and Coordinates. See Fig. 10.8
(3) Formulation 33

34. (ii) Governing Equations. Obtained from eq. (10.8)
(i) Assumptions
(1) All assumptions leading to eqs. (10.8) and (10.9) are applicable
(2) Steady state
(3) One-dimensional
(4) T
issue temperature at the base x
= 0 is equal to
(5) Skin is maintained at uniform temperature
(6) Negligible blood perfusion in the cutaneous layer.
(ii) Governing Equations. Obtained from eq. (10.8)
(a) 34

35. (4) Solution (iii) Boundary Conditions (d) ) ( T = T L = ) ( (e)) ( a T = s T L = ) (
(e)
(4) Solution
Define
Substituting (b), (c) and (f) into (a)
Boundary conditions 35

36. [ ] ò ò 1 ) ( = q (h) ) 1 ( = q (i) Integrating (g) once gx x q - = +( = q
(h) ) 1 ( = q
(i)
Integrating (g) once [ ] gx x q - = + 1 2 ) ( C d B A Pe
integrating again 2 1 ) ( C B A Pe d + - = ò x g q
(j)
integrals (j) are of the form ò + 2 x c b a d
and
(k)
where 36

37. Evaluate integrals, substitute into (j)1 2
APe a + =
BPe b
CPe c
(m)
Evaluate integrals, substitute into (j) 2 1
tan ) ( ln C d c b a + ú û ù ê ë é - = x g q
(n) 2 4 b ac d - =
(o)
Boundary conditions (h) and (i) give the
constants 1 C
and 2 37

38. where Note: (1) a , b c and d depend on Listed in Table 10.1 (2)Pe
Listed in Table 10.1
(2) 1 C
depends on both o Pe
and : g 38

39. 39

40. (5) Checking (6) Comments Dimensional check:
Boundary conditions (h) and (i)
are satisfied
Boundary conditions check:
Tissue temperature increases as blood
perfusion and metabolic heat are increased
Qualitative check:
(6) Comments
(i) Enhancement in
eff k
due to blood perfusion
(ii) Temperature distribution for 60 = Pe
and 02 . g is
nearly linear. At
180 6
the
temperature is h
igher 40

41. 10.4. 5 The s-Vessel Tissue Cylinder Model
(iii) The governing parameters are Pe
and g
. The two are
physiologically related
(iv) Neglecting blood perfusion in the cutaneous layer
during vigorous exercise is not reasonable
The s-Vessel Tissue Cylinder Model
Model Motivation
Shortcomings of the Pennes equation
The Chen-Holmes equation and the Weinbaum-Jiji equation are
complex and require vascular geometry data
(a) Basic Vascular Unit
Vascular geometry of skeletal muscles has common features
Main supply artery and vein, SAV 41

42. Terminal arterioles and venules, t Capillary beds, c
Primary pairs, P
Secondary pairs, s
Terminal arterioles and venules, t
Capillary beds, c 42

43. Diameter and typically 10-15 mm long
NOTE: Blood flow in the SAV, P and s is countercurrent
Each countercurrent s pair is surrounded by a cylindrical tissue which is approximately 1 mm
Diameter and typically mm long
The tissue cylinder is a repetitive unit consisting of arterioles,
venules and capillary beds
This basic unit is found in most skeletal muscles
A bioheat equation for the cylinder represents the governing
equation for the aggregate of all muscle cylinders
(b) Assumptions
(1) Uniformly distributed blood bleed-off leaving artery is equal to that returning to vein of the s vessel pair 43

44. (c) Formulation Capillaries, arterioles and venules are essentially in
(2) Negligible axial conduction through vessels and cylinder
(3) Radii of the s vessels do not vary along cylinder
(4) Negligible temperature change between inlet to P vessels and
inlet to the tissue cylinder
(5) Temperature field in cylinder is based on conduction with a
heat-source pair representing the s vessels
(6) Outer surface of cylinder is at uniform temperature
(c) Formulation
Capillaries, arterioles and venules are essentially in
local thermal equilibrium with the surrounding tissue
Three temperature variables are needed: T , a
and v
Three governing equations are formulated 44

45. Navier-Stokes equations of motion give the velocity field
in the s vessels (axially changing Poiseuille flow)
Boundary Conditions
(1)    Continuity of temperature at the surfaces of the vessels
(2)    Continuity of radial flux at the surfaces of the vessels 45

46. (d) Solution The three eqs. for T , and are solved analytically
Solution gives vb T
, the outlet bulk vein temperature at = x
Simplified Case
Assume: 46

47. Artery and vein are equal in size
(1)
Artery and vein are equal in size
(2) Symmetrically positioned relative
to center of cylinder, i.e., v a l =
Results 47

48. (e) Modification of Pennes Perfusion Term
Eq. (10.18) gives
(a) into (10.21)
Dividing by the volume of cylinder 48

49. Blood flow energy generation per unit tissue volume:
(10.23) and 10.24) into (10.22)
(10.25) becomes 49

50. Artery supply temperature (1) body core temperature Ta T
Use (10.26) to replace the blood perfusion term in the
Pennes equation (10.3)
NOTE:
(1) This is the bioheat equation for the s-vessel cylinder
model 50

51. (2) is a correction coefficient defined in (10.18)
(a) It depends only on the vascular geometry of the tissue
cylinder
(b) It is independent of blood flow rate
(c) Its value for most muscle tissues ranges from 0.6 to 0.8
(d) This vascular structure parameter is much simpler
than that required by Chen-Holmes and Weinbaum-
Jiji equations
(3) The model analytically determines the venous return
temperature
(4) Accounts for contribution of countercurrent heat
exchange in the thermally significant vessels. 51

52. Example 10.3: Surface Heat Loss from Peripheral Tissue
(a)        It is approximated by the body core temperature in
the Pennes bioheat equation
(b)        Its determination involves countercurrent heat
exchange in SAV vessels
(6)    While equations (10.5) and (10.6) apply to the cutaneous
layer of peripheral tissue, eq applies to the region
below the cutaneous layer.
Example 10.3: Surface Heat Loss from Peripheral Tissue
Peripheral tissue of thickness L 52

53. (2) Origin and Coordinates. See Fig. 10.12
Cutaneous plexus:
Perfusion rate bc w
&
(uniformly distributed),
blood supply temperature cb T
Skin temperature s T
Metabolic heat
Specified correction coefficient * T D
Use the s-vessel tissue cylinder model, determine surface
flux
(1)      Observations
Temperature distribution gives surface flux
This is a two layer problem: tissue and cutaneous
(2) Origin and Coordinates. See Fig 53

54. (ii) Governing Equations
((3) Formulation
(i)     Assumptions
(1) Apply all assumptions leading to (10.5) and
(10.27)
(2) Steady state
(2) One-dimensional
(3) Constant properties
(4) Uniform metabolic heat in tissue layer
(5) Negligible metabolic heat in cutaneous layer
(7) Uniform blood perfusion in cutaneous layer
(ii) Governing Equations
Fourier’s law at surface: 54

55. (iii) Boundary Conditions
Need 2 equations: one for tissue layer and one for cutaneous
Tissue layer temperature T: eq. (10.27):
(iii) Boundary Conditions 55

56. (4) Solution
Let
(b) and (c) become 56

57. Dimensionless parameters:
(k) into (i) and (j) 57

58. Solutions to (m) and (n):
Boundary conditions
Solutions to (m) and (n):
Boundary conditions (p)-(s) give constants 58

59. 59

60. Surface heat flux:
(5) Checking
Dimensional check:
Limiting check: 60

61. (5) Comments 61